Triangle calculator

You have entered side a, b and angle β.

Triangle has two solutions: a=38.5; b=34.5; c=5.14552145419 and a=38.5; b=34.5; c=56.75217637256.

#1 Obtuse scalene triangle.

Sides: a = 38.5 b = 34.5 c = 5.14552145419

Area: T = 58.91444489058
Perimeter: p = 78.14552145419
Semiperimeter: s = 39.0732607271

Angle ∠ A = α = 138.4110608231° = 138°24'38″ = 2.41657208333 rad
Angle ∠ B = β = 36.5° = 36°30' = 0.6377045177 rad
Angle ∠ C = γ = 5.0899391769° = 5°5'22″ = 0.08988266433 rad

Height: h_a = 3.06604908523
Height: h_b = 3.41553303714
Height: h_c = 22.90106772899

Median: m_a = 15.42107365693
Median: m_b = 21.37328593394
Median: m_c = 36.46441151247

Inradius: r = 1.50878197495
Circumradius: R = 299.0002339927

Vertex coordinates: A[5.14552145419; 0] B[0; 0] C[30.94884891338; 22.90106772899]
Centroid: CG[12.03112345586; 7.63435590966]
Coordinates of the circumscribed circle: U[2.5732607271; 28.88659007729]
Coordinates of the inscribed circle: I[4.5732607271; 1.50878197495]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 41.5899391769° = 41°35'22″ = 2.41657208333 rad
∠ B' = β' = 143.5° = 143°30' = 0.6377045177 rad
∠ C' = γ' = 174.9110608231° = 174°54'38″ = 0.08988266433 rad

How did we calculate this triangle?

1. Input data entered: side a, b and angle β.

$a = 38.5 ; ; b = 34.5 ; ; beta = 36.5° ; ;$

2. From angle β, side a and side b we calculate side c - by using the law of cosines and quadratic equation:

$b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 34.5**2 = 38.5**2 + c**2 - 2 * 38.5 * c * cos 36° 30' ; ; ; ; ; ; c**2 -61.897c +292 =0 ; ; a=1; b=-61.897; c=292 ; ; D = b**2 - 4ac = 61.897**2 - 4 * 1 * 292 = 2663.23591865 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 61.9 ± sqrt{ 2663.24 } }{ 2 } ; ; c_{1,2} = 30.94848913 ± 25.8032745919 ; ; c_{1} = 56.7517637219 ; ; c_{2} = 5.14521453815 ; ;$
$; ; text{ Factored form: } ; ; (c -56.7517637219) (c -5.14521453815) = 0 ; ; ; ; c > 0 ; ; ; ; c = 56.752 ; ;$

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

$a = 38.5 ; ; b = 34.5 ; ; c = 5.15 ; ;$

3. The triangle circumference is the sum of the lengths of its three sides

$p = a+b+c = 38.5+34.5+5.15 = 78.15 ; ;$

4. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 78.15 }{ 2 } = 39.07 ; ;$

5. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39.07 * (39.07-38.5)(39.07-34.5)(39.07-5.15) } ; ; T = sqrt{ 3470.91 } = 58.91 ; ;$

6. Calculate the heights of the triangle from its area.

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 58.91 }{ 38.5 } = 3.06 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 58.91 }{ 34.5 } = 3.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 58.91 }{ 5.15 } = 22.9 ; ;$

7. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 34.5**2+5.15**2-38.5**2 }{ 2 * 34.5 * 5.15 } ) = 138° 24'38" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 38.5**2+5.15**2-34.5**2 }{ 2 * 38.5 * 5.15 } ) = 36° 30' ; ; gamma = 180° - alpha - beta = 180° - 138° 24'38" - 36° 30' = 5° 5'22" ; ;$

8. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 58.91 }{ 39.07 } = 1.51 ; ;$

9. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 38.5 }{ 2 * sin 138° 24'38" } = 29 ; ;$

10. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.5**2+2 * 5.15**2 - 38.5**2 } }{ 2 } = 15.421 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.15**2+2 * 38.5**2 - 34.5**2 } }{ 2 } = 21.373 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.5**2+2 * 38.5**2 - 5.15**2 } }{ 2 } = 36.464 ; ;$

#2 Obtuse scalene triangle.

Sides: a = 38.5 b = 34.5 c = 56.75217637256

Area: T = 649.8276913357
Perimeter: p = 129.7521763726
Semiperimeter: s = 64.87658818628

Angle ∠ A = α = 41.5899391769° = 41°35'22″ = 0.72658718203 rad
Angle ∠ B = β = 36.5° = 36°30' = 0.6377045177 rad
Angle ∠ C = γ = 101.9110608231° = 101°54'38″ = 1.77986756563 rad

Height: h_a = 33.75772422523
Height: h_b = 37.6711125412
Height: h_c = 22.90106772899

Median: m_a = 42.83662445014
Median: m_b = 45.32204572239
Median: m_c = 23.04547245266

Inradius: r = 10.01664636641
Circumradius: R = 299.0002339927

Vertex coordinates: A[56.75217637256; 0] B[0; 0] C[30.94884891338; 22.90106772899]
Centroid: CG[29.23334176198; 7.63435590966]
Coordinates of the circumscribed circle: U[28.37658818628; -5.9855223483]
Coordinates of the inscribed circle: I[30.37658818628; 10.01664636641]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138.4110608231° = 138°24'38″ = 0.72658718203 rad
∠ B' = β' = 143.5° = 143°30' = 0.6377045177 rad
∠ C' = γ' = 78.0899391769° = 78°5'22″ = 1.77986756563 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and angle β.

$a = 38.5 ; ; b = 34.5 ; ; beta = 36.5° ; ; : Nr. 1$

2. From angle β, side a and side b we calculate side c - by using the law of cosines and quadratic equation:

$b**2 = a**2 + c**2 - 2a c cos beta ; ; ; ; 34.5**2 = 38.5**2 + c**2 - 2 * 38.5 * c * cos 36° 30' ; ; ; ; ; ; c**2 -61.897c +292 =0 ; ; a=1; b=-61.897; c=292 ; ; D = b**2 - 4ac = 61.897**2 - 4 * 1 * 292 = 2663.23591865 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -b ± sqrt{ D } }{ 2a } = fraction{ 61.9 ± sqrt{ 2663.24 } }{ 2 } ; ; c_{1,2} = 30.94848913 ± 25.8032745919 ; ; c_{1} = 56.7517637219 ; ; c_{2} = 5.14521453815 ; ; : Nr. 1$
$; ; text{ Factored form: } ; ; (c -56.7517637219) (c -5.14521453815) = 0 ; ; ; ; c > 0 ; ; ; ; c = 56.752 ; ; : Nr. 1$

Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

$a = 38.5 ; ; b = 34.5 ; ; c = 56.75 ; ;$

3. The triangle circumference is the sum of the lengths of its three sides

$p = a+b+c = 38.5+34.5+56.75 = 129.75 ; ;$

4. Semiperimeter of the triangle

$s = fraction{ o }{ 2 } = fraction{ 129.75 }{ 2 } = 64.88 ; ;$

5. The triangle area using Heron's formula

$T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 64.88 * (64.88-38.5)(64.88-34.5)(64.88-56.75) } ; ; T = sqrt{ 422275.02 } = 649.83 ; ;$

6. Calculate the heights of the triangle from its area.

$T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 649.83 }{ 38.5 } = 33.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 649.83 }{ 34.5 } = 37.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 649.83 }{ 56.75 } = 22.9 ; ;$

7. Calculation of the inner angles of the triangle using a Law of Cosines

$a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 34.5**2+56.75**2-38.5**2 }{ 2 * 34.5 * 56.75 } ) = 41° 35'22" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 38.5**2+56.75**2-34.5**2 }{ 2 * 38.5 * 56.75 } ) = 36° 30' ; ; gamma = 180° - alpha - beta = 180° - 41° 35'22" - 36° 30' = 101° 54'38" ; ;$

8. Inradius

$T = rs ; ; r = fraction{ T }{ s } = fraction{ 649.83 }{ 64.88 } = 10.02 ; ;$

9. Circumradius

$R = fraction{ a }{ 2 * sin alpha } = fraction{ 38.5 }{ 2 * sin 41° 35'22" } = 29 ; ;$

10. Calculation of medians

$m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.5**2+2 * 56.75**2 - 38.5**2 } }{ 2 } = 42.836 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 56.75**2+2 * 38.5**2 - 34.5**2 } }{ 2 } = 45.32 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 34.5**2+2 * 38.5**2 - 56.75**2 } }{ 2 } = 23.045 ; ;$
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